Problem 12 - Elementary Algebra Within Reach

Question

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{rr} x+6 y & =7 \\ -x+4 y & =-2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -8\) and \(y = 2.5\).

Step 1: Solve first equation for one variable

To use the substitution method, one of the equations needs to be solved for one of the variables. The first equation is \(x+6y = 7\). Solving this for \(x\) gives \(x = 7-6y\).

Step 2: Substitute the result from Step 1 into the second equation

Now replace \(x\) in the second equation \(-x+4y = -2\) with the expression \(7-6y\) from step 1: \(-(7-6y)+4y = -2\).

Step 3: Solve the resulting equation for y

Combine like terms to simplify the equation from step 2: \(-7+2y = -2\). Now solve this for \(y\) to get \(y=2.5\).

Step 4: Substitute y value back into first equation to solve for x

With \(y = 2.5\), substitute this value back into the first equation from step 1 to solve for \(x\): \(x = 7-6(2.5) = -8\).

Key Concepts

System of Equations

A system of equations is essentially a set of two or more equations that share the same variables. We aim to find values for these variables that satisfy each equation concurrently. In our exercise, we have a system with two linear equations:

• \(x + 6y = 7\)
• -\(x + 4y = -2\)

We are tasked with finding the values of \(x\) and \(y\) that make both equations true at the same time. Using systems of equations is very common in algebraic problems, as it allows us to deal with multiple conditions using the same set of variables. Each equation represents a different relationship between those variables, and solving the system means finding a balance that satisfies all these relationships.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. Each equation forms a straight line when graphed, which is why these are called linear equations. When handling a single linear equation like \(x+6y=7\), the goal is to isolate one of the variables to determine its value. For instance, in the example given, we first solve for \(x\) in terms of \(y\) by rearranging the equation as \(x = 7 - 6y\). After obtaining this expression, we substitute it into the other equation. This substitution transforms our system into a single equation with only one variable, which simplifies the process of finding a solution. The final step involves substituting back to verify and find the value of the other variable.

Algebraic Methods

The substitution method is one of the core algebraic techniques used to solve systems of equations. By rearranging one equation to isolate a variable, substitution simplifies the solution process by reducing the number of variables you contend with in a given equation. Here's a simplified approach to using this method:

• Choose one of the equations and solve for one variable in terms of the other.
• Use this expression to replace the same variable in the other equation.
• Solve the resulting single-variable equation.
• Substitute the found value back to find the remaining variable.

This method is particularly beneficial when one of the equations is easy to manipulate. Employing algebraic methods like substitution can make solving complex systems much more approachable.